Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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37 views

Discriminate closed-chain from open-chain

I have two classes of chains: closed-chains where a random path ends near where it starts (ie. loop), and open-chains without this restriction (ie. random walk). These chains are a directed graph ...
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2answers
64 views

Transitive Closure of Spacelike Separation

Let $S$ be a set of (possibly infinitely many) events in Minkowski spacetime. What would be the necessary and sufficient condition for $S$ (or the elements of S) to be such that for any $x, y, z$ $\in ...
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Is space-like separation transitive?

Suppose that events $A$ and $B$ are spacelike separated. Also suppose that events $B$ and $C$ are spacelike separated. Does this guarantee that $A$ and $C$ are spacelike separated? That is, is the ...
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60 views

Explain why the universe could be compact

Regarding the topology of the universe, it could be compact like a sphere or open like a Euclidean space, but since the universe started from a single point, doesn't that mean that the shape of the ...
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1answer
25 views

Plotting a Fibonacci Spiral Simultaneously in 3-dimensions [closed]

I have a question today that's to do with the Fibonacci (Golden) Spiral... How would you plot the following graph? The Fibonacci Spiral, but as the spiral moves in the x-y plane, the spiral is also ...
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1answer
82 views

Time-reversal symmetry for spin Hamiltonian

In the topology online course by TU Delft, the time-reversal operator acting on a system of spin-1/2 particles is introduced as $$ \mathcal T = i\sigma_y\mathcal K. $$ I understand this acts on the ...
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21 views

An instanton in $d$ dimensions is often a soliton in $d + 1$ dimensions?

The title of this questions is a "folklore" I've heard from a lot of researchers, but I never understood why this is the case. I know what an instanton and soliton is, respectively in the ...
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71 views

Are topological changes to dynamic spacetime quantized? Can the Chern-Gauss-Bonnett theorem illuminate dynamics?

I was looking at the Chern-Gauss-Bonnett theorem in dimension 4. Here we can write the Euler characteristic of a compact 4-manifold as: $$\chi(M)=\frac{1}{32\pi^{2}}\intop_{M}\left(|\mathrm{Riem}|^{2}-...
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76 views

Spin connection for a paralellization takes more general forms than $SO(3,1)$ in different spacetime topologies?

I'm interested in a frame bundle over spacetimes with different topologies. In the trivial case of Minkowskian space ($\mathbb{R}^{3,1}$), a frame (or tangent space) at one point is going to be ...
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Why does a choice of this $\psi$ in the worldsheet metric corresponds to a choice of complex structure?

As far as I'm aware, a complex manifold $M$ is a topological manifold together with an atlas ${\cal A}$ of charts $(U_i,\varphi_i)\in{\cal A}$ such that the open sets $U_i$ cover $M$, the maps $\...
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4answers
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Do objects in a 2D universe have an edge?

When discussing a 2D universe, many assume that an object would only be seen as "a line". This would imply that you are seeing the "edge" of the object. But, if there are only ...
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Isometric embedding of embedding of Schwarzchild metric [duplicate]

I am reading through this article https://arxiv.org/abs/1010.4256 about the special case of the positive mass theorem in general relativity. I do not understand the section below: In particular what ...
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Why the work done in a conservative field around a closed circle does not vanish when calculated in cylindrical coordinates?

I was solving problem 2.4.13 from the book "George B Arfken, Hans J Weber - Mathematical Methods For Physicists- Sixth edition" and the problems was that: Problem 2.4.13 A force is ...
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54 views

Isometric embedding of Schwarzchild metric in $\mathbb{R}^4$

I am reading through this article https://arxiv.org/abs/1010.4256 about the special case of the positive mass theorem in general relativity. I do not understand the section below: In particular what ...
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20 views

Trivial examples for the Chern number from the potential for quantized transport

I'm trying to understand the phenomena of quantized electron transport better. The difficult step is that for any Hamiltonian (where $V(x,t)$ is periodic in both arguments and is a slow function of $t$...
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Magnetic monopole, vector-potential and differential forms

When written in the language of exterior algebra, Maxwell-Thomson equation writes as $dB=0$ where $d$ is the exterior derivative and $B$ is the magnetic flux 2-form. From Poincaré's lemma, it follows ...
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21 views

The volume enclosing the charge in Gauss's law: does it have to be simply connected?

I was trying to apply Gauss's law to a simple problem: Find the capacitance of a cylindrical capacitor. Inner radius is $a$ and outer radius is $c$. The space between the plates is a dielectric ...
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81 views

Connectedness in phase-space

In my statistical mechanics lecture, it was claimed that a volume of phase-space cannot be split into two separate volumes as time evolves. I suspect that this is a topological fact that I am not ...
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35 views

Is there physical reason for a stably causal spacetime, or the existence of a Cauchy surface?

In their 1979 essay Global structure of spacetimes, Geroch and Horowitz describe methods of determining the topology, causal structure and singularity of spacetimes. Their (mathematical) arguments are ...
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1answer
59 views

Is it enough to give a time-orientation to define a spin structure?

Maybe I got it wrong and my question doesn't make sense, excuse me if that's the case. For a smooth Lorentz 4-manifold $(M, g)$ with signature $(- + + +)$ is it enough to give a time-orientation to ...
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lecture notes about the relation between algebraic topology, topological quantum field theory, condensed matter physics [closed]

I am an undergraduate student and I am very interested in topology with its application in physics. So last year I've read some books about this field, mainly about topological soliton, some ...
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Can magnetic loops with no source current knot or link?

The answer to this question is obviously no. I would like to pose a variation of that question. Suppose a simply connected domain of a 3-d vacuum space has no source current. Does there exist a case ...
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1answer
656 views

Magnetic field loops do not knot or link

The magnetic field is composed of closed loops (assuming there is no magnetic monopole). How does one prove any two magnetic loops do not knot to form a link?
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2answers
98 views

Does a positive curvature necessarily indicate the finiteness of the universe?

Imagine the following situation: more and more accurate measurements of the average density of the Universe reveal that it is greater than the critical one, which corresponds to the model of a closed ...
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59 views

About Chern Insulator

I know when we talk about Insulator, U(1)charge symmetry naturally exists. But in this article:Quantum phase transitions of topological insulators without gap closing, the author claims that: "...
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1answer
50 views

Spin 1/2 as belt trick in a smooth field

In the (English) Wikipedia article on Spinor, there is an animation, demonstrating the Dirac belt trick as a model for Spin 1/2. My interpretation of that animation goes like this: If you rotate an ...
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1answer
34 views

If the universe had a topological hole, would moving around the hole have a centripetal force?

I understand that if it were of a toroidal topology, it would not literally mean that the universe is in the shape of a 3D donut. However, I can't seem to draw intuition on why or why not it may be ...
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45 views

Does the fiber bundle approach for Berry connection contradict adiabatic theorem?

In Ref [1], the authors show how the Berry connection is a geometric quantity using the fiber bundle approach. My question is about the idea of taking a local section of a fiber bundle (corresponding ...
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1answer
75 views

How can a classical phase space be unquantizable?

On page 2 of the paper "2 + 1 dimensional gravity as an exactly soluble system" Witten claims that: Depending on its topology, a finite-dimensional phase space might be unquantizable, How ...
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1answer
41 views

What are the two different $\mathbb{S}^n$ in the construction of the homotopy group $\pi_n(\mathbb{S}^n)$ that classifies topological defects?

According to Mukhanov's Physical Foundations of Cosmology, Homotopy groups give us a useful unifying description of topological defects. Maps of the $n$-dimensional sphere $\mathbb{S}^n$ into a ...
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13 views

Why linear optical response is absent in a non-centrosymmetric system with time reversal symmetry?

In this paper, it is mentioned that a non-centrosymmetric system with time-reversal symmetry doesn't have a linear response. It is actually a consequence of the Onsager reciprocal theorem. But I didn'...
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1answer
37 views

What closed 3D space looks and behaves like? (Relativistic Black Hole)

So I wanted to ask a question that is a copy of Why can't you escape a black hole? From the answers, the conclusion I draw is: it's impossible to escape a black hole. any trajectory inside the ...
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2answers
88 views

Non-Minkowskian spacetime with cancelling Riemann tensor

I recently read that (at least in $2+1$ dimensions but maybe it's true in general) the fact that all the component of Riemann tensor are identically 0: \begin{equation} R_{\alpha\beta\mu\nu} = 0, \end{...
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1answer
66 views

Spacetimes with “celestial Riemann surface” other than the sphere

In the standard study of asymptotically flat spacetimes one defines null infinity demanding that topologically ${\cal I}^\pm \simeq \mathbb{R}\times S^2$ (c.f. Definition 1 of this review by Ashtekar)....
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1answer
52 views

Black hole atlases

What sort of atlases of spaces that contain a black hole (that is, including the space inside the event horizon), if any, are there? Does the central singularity have to be excluded? Are there atlases ...
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85 views

Equipotential surfaces for a knotted charge distribution

Suppose we have a compact submanifold $K$ of $\mathbb{R}^3$ with uniformly distributed charge. Neglecting multiplicative constants, the electric potential $\Phi(\vec{x}) = \int_{K} \frac{dK}{|x - k|}$...
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24 views

Why is the ground state degeneracy of the toric code 4?

Hi I'm kind of confused about the ground state degeneracy in the toric code model. The generic ground state of the TCM is a state $|\Omega\rangle$: $A_v |\Omega \rangle = B_p | \Omega \rangle = | \...
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1answer
110 views

A doubt on the Topology of Einstein-Rosen Bridges (or Schwarzschild/Kruskal Wormholes)

Well, one of the "mantras" of General Relativity is: Einstein Field Equations concerns about the local geometrical structure of spacetime (the metric tensor) and tell you nothing about the ...
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53 views

About the $\sigma_{xy}$ in the integer quantum Hall effect (or quantum anomalous Hall effect)

We know that $\sigma_{xy}$ in the integer quantum Hall effect (or quantum anomalous Hall effect) can be calculated by the Berry curvature, but we also know that $\sigma_{xy}$ is calculated by the ...
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Relating the topological behaviour in the toric code to cohomology?

I've been working on the Toric Code Model (by Kitaev in his 2003 paper on quantum computation), and the model is a lattice realisation of a topological phase. The local operators in the model are ...
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1answer
75 views

Open sets in Minkowski spacetime

I don't know how to imagine open sets in Minkowski spacetime. I have seen that there are many diffrent ways of constructing them — that's OK. But for example. which construction do people mean in the ...
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1answer
44 views

What's the index (or topological charge) of this vector field image?

I am doing some research in a condensed matter system, and found this Berry curvature / vector field configuration that is unusual. I cannot find another example of something similar, either from ...
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41 views

How does a lattice regulator work if we don't want observables to be invariant under “large” gauge transformations?

In quantum field theory (QFT), observables must be invariant under gauge transformations that are continuously connected to the identity, but invariance under "large" gauge transformations ...
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How is topology related to physics?

Topology has many occurences in physics like topological insulators, topological quantum computing etc. But what is confusing me is that topology is this mathematical theory that studies the behaviour ...
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78 views

Theoretically, how could a wormhole be made? [duplicate]

Concentrating a lot of matter in one place will make a black hole, not a wormhole. A wormhole would change the topology of spacetime. Does General Relativity allow this? I know there are wormhole ...
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25 views

Deforming a nematic line defect to a uniform configuration

In Nakahara section 4.9, "Defects in nematic liquid crystals", it is discussed that the order parameter for a nematic should be the real projective plane $\mathbb{R}P^2$, which has ...
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How does the Bloch sphere indicate topology of 2-level $k\cdot p$ effective Hamiltonians?

It is known that the topology of some parameter space of a 2-level system (such as the Brillouin torus) may be found via the Gauss map to the Bloch sphere. The topology is indicated by the number of ...
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1answer
88 views

What is topological in Kitaev Chain

What is topological in Kitaev Chain? Realspace or the space of Pauli spins or the space of fermions? My Understanding I understand that majorana-zero modes are which are spatially separated, are ...
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30 views

Cosmic strings, monopoles and textures

I am a beginner in topology and I want some help in visualizing the following statements: "Strings arise when manifold contains non-contractible loops" "Monopoles arise when manifold ...
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1answer
110 views

Does spacetime topology have importance in physics?

Generally in textbooks they represent spacetime as $(M,\nabla,g,t)$ where $M$ is a Lorentzian manifold,$\nabla$ a torsion-free connection,$g$ a metric and $t$ a time orientation. But they do not talk ...

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